Consider a hierarchical log-linear model, given by a simplicial complex, $Γ$, and integer matrix $A_Γ$. We give a new characterization of the rank of $A_Γ$ given by a logarithmic transformation on the exponential Hilbert series of $Γ$. We show that, if each random variable in $X$ has the same number of possible outcomes, then this formula reduces to a simple description in terms of the face vector of $Γ$. If $Γ$ further satisfies the Dehn-Sommerville relations, then we give an exceptionally simple formula for computing the rank of $A_Γ$, and thus the dimension and the number of degrees of freedom of the model.